CHICAGO—Toyota Motor Corp. is taking another stab at breaking into the lucrative market for big pickup trucks, long the stronghold of its Detroit rivals.

Seven years ago, the Japanese auto maker built a $1.3 billion plant in Texas to produce the Tundra, a full-size pickup engineered to offer the might and toughness to go toe to toe with the likes of the Ford F-150 and Chevrolet Silverado.